Abstract
Vickrey (1991, 2020) proposed a bathtub model for the evolution of trip flows served by privately operated vehicles inside a road network based on three premises: (i) treatment of the road network as a single bathtub; (ii) the speed-density relation at the network level, also known as the network fundamental diagram of vehicular traffic, and (iii) the time-independent negative exponential distribution of trip distances. However, the distributions of trip distances are generally time-dependent in the real world, and Vickrey’s model leads to unreasonable results for other types of trip distance distributions. Thus there is a need to develop a bathtub model with more general trip distance distribution patterns.In this study, we present a unified framework for modeling network trip flows with general distributions of trip distances, including negative exponential, constant, and regularly sorting trip distances studied in the literature. In addition to tracking the number of active trips as in Vickrey’s model, this model also tracks the evolution of the distribution of active trips’ remaining distances. We derive four equivalent differential formulations from the network fundamental diagram and the conservation law of trips for the number of active trips with remaining distances not smaller than any value. Then we define and discuss the properties of stationary and gridlock states, derive the integral form of the bathtub model with the characteristic method, and present two numerical methods to solve the bathtub model based on the differential and integral forms respectively. We further study equivalent formulations and solutions for two special types of distributions of trip distances: time-independent negative exponential or deterministic. In particular, we present six equivalent conditions for Vickrey’s bathtub model to be applicable.
Highlights
The basic role of a transportation system is to serve trips of persons and goods through privately operated or mobility service vehicles
Vickrey (1991, 1994) introduced the first bathtub model for network trip flows served by privately operated vehicles based on three premises: (i) “a maze of congested streets is treated as an undifferentiated movement area”; (ii) “movement takes place at a speed which is a function of the density of cars in the area”; and (iii) the trip distance follows a time-independent negative exponential distribution, or the average remaining distance of active trips is constant
In this study we first presented a generalized bathtub model of network trip flow served by privately operated vehicles, in which a road network is treated as a single bathtub, and all vehicles’ speeds at a time instant are identical and determined by the network fundamental diagram
Summary
The basic role of a transportation system is to serve trips of persons and goods through privately operated or mobility service vehicles. In addition to vehicle densities, the distribution of trips can impact the average speed of mobility service vehicles due to the boarding and alighting times and other delays; and existing network fundamental diagrams usually ignore the impacts of dynamic trip flows It is a good starting point for studying traffic dynamics at the aggregate level for signalized and freeway networks. Vickrey (1991, 1994) introduced the first bathtub model for network trip flows served by privately operated vehicles based on three premises: (i) “a maze of congested streets is treated as an undifferentiated movement area”; (ii) “movement takes place at a speed which is a function of the density of cars in the area”; and (iii) the trip distance follows a time-independent negative exponential distribution, or the average remaining distance of active trips is constant.
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